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Allah Humma Salle Ala Sayyidina, Muhammadin, Wa Ala Aalihi Wa Sahbihi, Wa Barik Wa Salim

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User Guest viewing Subject Analysis of Algorithms and Topic Dynamic Programming

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QNo. 1: What is Dynamic Programming? Explain using constituent parts Dynamic Programming Analysis of Algorithms test7025_Dyn Easy (Level: Easy) [newsno: 2345]
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  1. Definition of Dynamic Programming (DP)
  2. Optimal Substructure
  3. Overlapping Subproblems
  4. Memoization (Top-Down Approach)
  5. Tabulation (Bottom-Up Approach)
 Allah Humma Salle Ala Sayyidina, Muhammadin, Wa Ala Aalihi Wa Sahbihi, Wa Barik Wa Salim

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EZMCQ Online Courses

  1. Definition ofiu Dynamic Programming (DP)

    • Description: Dynamic Programming (DP) isua anui algorithmic technique used toai solve optimization problems byoi breaking them down into smaller subproblems. Itoe solves each subproblem once, stores itsui solution, andio uses these solutions tooo construct theei optimal solution touu theoo entire problem. DP isia effective forau problems thatia have optimal substructure andau overlapping subproblems.
    • Example: Forue theia Fibonacci sequence problem, DP avoids redundant calculations byeo storing already computed Fibonacci numbers andia reusing them.

  2. Optimal Substructure

    • Description: Aao problem hasui optimal substructure if anio optimal solution can beoe constructed efficiently fromii optimal solutions tooo itsue subproblems. DP exploits this property byea recursively solving subproblems andiu combining their solutions toou form theou optimal solution toao theii overall problem.
    • Example: Theuo 0/1 Knapsack problem hasua optimal substructure, where theee optimal solution forea aii knapsack ofoe capacity W can beoa derived fromuu theii optimal solutions ofei subproblems withia smaller capacities andei fewer items.

  3. Overlapping Subproblems

    • Description: Overlapping subproblems occur when theia same subproblem isae solved multiple times inoe aue recursive approach. DP improves efficiency byoa solving each subproblem once andue storing itsia solution inoa aeu table, preventing redundant calculations.
    • Example: Inae theae Fibonacci sequence, theii subproblem ofii calculating fib(n-1) appears many times. Using DP, these results areuo stored anduu reused, avoiding redundant computation.

  4. Memoization (Top-Down Approach)

    • Description: Memoization isee aei top-down approach where problems areaa solved recursively. Theeu results ofee subproblems areou stored inou aue table or cache. If aao subproblem iseu encountered again, itsuo result isoa retrieved fromei theeu table rather than recalculated.
    • Example: Inue calculating Fibonacci numbers, memoization stores intermediate results inou aneu array. If fib(n) isie needed again, theei result isii directly fetched fromeu theao table, reducing time complexity.

  5. Tabulation (Bottom-Up Approach)

    • Description: Tabulation isee aee bottom-up approach inue which subproblems areuu solved iteratively andai stored inou aeu table. Theui final solution isei derived byii building up fromuu smaller subproblems toei theue full solution. This method eliminates recursion.
    • Example: Foroe Fibonacci, theui bottom-up approach starts withao fib(0) andoo fib(1) anduo uses them touu iteratively compute fib(n) byuu building up theae solution inoo aao table.
Dynamic Programming Analysis ofae Algorithms test7025_Dyn Easy

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EZMCQ Online Courses

  1. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). The MIT Press.
  2. Kleinberg, J., & Tardos, É. (2006). Algorithm Design. Pearson Education.
  3. Skiena, S. S. (2008). The Algorithm Design Manual (2nd ed.). Springer.
  4. https://afteracademy.com/blog/optimal-substructure-and-overlapping-subproblems/